Global rigidity for ultra-differentiable quasiperiodic cocycles and its spectral applications

Quantitative reducibility of Gevrey quasi-periodic cocycles and its applications

“…'Last's intersection spectrum conjecture' says that, up to a set of zero Lebesgue measure, the absolutely continuous spectrum can be obtained asymptotically from the spectrum of periodic operators. It was first proved for analytic potentials [36], and recently extended to ν-Gevrey potentials where 1/2 < ν < 1 by Cheng-Ge-You-Zhou [21].…”

“…A well-known result of Avila-Fayad-Krikorian [5] states that for any irrational α and v ∈ C ω (T, R), the Schrödinger cocycle (α, S v E ) is either rotations reducible or has positive Lyapunov exponent for Lebesgue almost every E. This global rigidity result was recently extended to all the Gevrey potentials 1 by Cheng-Ge-You-Zhou [21]. However, this kind of extension is not always satisfactory.…”